Finite Group Databasebeta

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#72245418

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#152323
commutator#532C22×C4
Frattini, cosocle#3151C23
prime radical#60798C22
LCSG ▷ #532 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C22×C42 ◁ #60798 ◁ G
LECSG ▷ #3151 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

745 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 745 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,4,10,20,12,5,11,6,9,19,28,7,22,21,29)(3,13,15,25,32,24,17,8,14,23,16,26,18,30,31,27);(1,3,7,18,6,14,20,32)(2,8,22,26,9,27,12,25)(4,15,5,17,19,16,21,31)(10,24,11,13,28,30,29,23);(2,9)(3,14)(4,19)(5,21)(10,28)(11,29)(12,22)(18,32)

Polycyclic presentation:

2^10|a^2=d,b^2=dfhi,d^2=hij,f^2=hij,g^2=i,h^2=j,b^a=be,c^a=cdfj,c^b=cj,d^b=dg,d^c=dij,e^a=egi,e^b=eij,e^c=eij,e^d=ehij,f^a=fij,f^b=fg,f^c=fij,f^d=fj,f^e=fhi,g^a=gh,g^b=gij,g^e=gi,h^a=hj,h^b=hi,h^e=hj,i^a=ij

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245425
maximal normal subgroup of#72245425

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.6*F.8*F.9);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.6*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);